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Topic 1.2.6Topic 1.2.6

Order of OperationsOrder of Operations

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Topic1.2.6

Order of OperationsOrder of Operations

California Standard:1.1 Students use properties of numbers to demonstrate whether assertions are true or false.

What it means for you:You’ll see why it’s important to have a set of rules about the order in which you have to deal with operations.

Key words:• grouping symbols• parentheses• brackets• braces• exponents

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Topic1.2.6

Order of OperationsOrder of Operations

It’s important that all mathematicians write out expressions in the same way, so that anyone can reach the same solution by following a set of rules called the “order of operations.”

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Topic1.2.6

Grouping Symbols Show You What to Work Out First

Order of OperationsOrder of Operations

If you wanted to write a numerical expression representing

“add 4 and 3, then multiply the answer by 2,”

you might be tempted to write 4 + 3 × 2.

But watch out — this expression contains an addition and a multiplication, and you get different answers depending on which you do first.

If you do the addition first, you get the answer 7 × 2 = 14.

If you do the multiplication first, you get the answer 4 + 6 = 10.

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Topic1.2.6

Order of OperationsOrder of Operations

You might know the addition has to be done first, but somebody else might not.

To be really clear which parts of a calculation have to be done first, you can use grouping symbols.

parentheses ( ),

brackets [ ],

and braces { }.

Some common grouping symbols are:

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Topic1.2.6

Example 1

Solution follows…

Order of OperationsOrder of Operations

Write an expression representing the phrase “add 4 and 3, then multiply the answer by 2.”

Solution

You need to show that the addition should be done first, so put that part inside grouping symbols:

The expression should be (4 + 3) × 2.

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Topic1.2.6

Guided Practice

Solution follows…

Order of OperationsOrder of Operations

Write numeric expressions for these phrases:

1. Divide 4 by 8 then add 3.

2. Divide 4 by the sum of 8 and 3.

3. From 20, subtract the product of 8 and 2.

4. From 20, subtract 8 and multiply by 2.

(4 ÷ 8) + 3

4 ÷ (8 + 3)

20 – (8 × 2)

(20 – 8) × 2

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Topic1.2.6

Guided Practice

Solution follows…

Order of OperationsOrder of Operations

Evaluate the following sums and differences:

5. (3 – 2) – 5 6. 3 – (2 – 5)

7. 6 – (11 + 7) 8. (7 – 8) – (–3 – 8) – 11

9. (5 – 9) – (3 – 10) – 2 10. 9 + (5 – 3) – 4

–4 6

–12 –1

1 7

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Topic1.2.6

Order of OperationsOrder of Operations

Nested grouping symbols are when you have grouping symbols inside other grouping symbols.

When you see nested grouping symbols, you always start from the inside and work outwards.

5 – [11 – (7 – 2)]

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Topic1.2.6

Example 2

Solution follows…

Order of OperationsOrder of Operations

Evaluate {5 – [11 – (7 – 2)]} + 34.

Solution

Start from the inside and work outwards:

{5 – [11 – (7 – 2)]} + 34 = {5 – [11 – 5]} + 34

= {5 – 6} + 34

= –1 + 34

= 33

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Topic1.2.6

Guided Practice

Solution follows…

Order of OperationsOrder of Operations

Evaluate the following:

11. 8 + [10 + (6 – 9) + 7]

12. 9 – {[(–4) + 10] + 7}

13. [(13 – 12) + 6] – (4 – 2)

14. 14 – {8 + [5 – (–2)]} – 6

15. 13 + [10 – (4 + 5)] – (11 + 8)

16. 10 + {[7 – (–2)] – (3 – 1)} + (–14)

8 + [10 + (–3) + 7] = 8 + 14 = 22

9 – {6 + 7} = 9 – 13 = –4

[1 + 6] – (4 – 2) = 7 – 2 = 5

14 – {8 + 7} – 6 = 14 – 15 – 6 = –7

13 + [10 – 9] – (11 + 8) = 13 + 1 – 19 = –5

10 + {9 – 2} + (–14) = 10 + 7 – 14 = 3

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Topic1.2.6

There are Other Rules About What to Evaluate First

Order of OperationsOrder of Operations

This order of operations is used by all mathematicians, so that every mathematician in the world evaluates expressions in the same way.

1. First calculate expressions within grouping symbols — working from the innermost grouping symbols to the outermost.

2. Then calculate expressions involving exponents.

3. Next do all multiplication and division, working from left to right.Multiplication and division have equal priority, so do them in the order they appear from left to right.

4. Lastly, do any addition or subtraction, again from left to right.Addition and subtraction have the same priority, so do them in the order they appear from left to right too.

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Topic1.2.6

Example 3

Solution follows…

Order of OperationsOrder of Operations

Simplify {4(10 – 3) + 32} × 5 – 11.

SolutionWork through the expression bit by bit: {4(10 – 3) + 32} × 5 – 11

= {4 × 7 + 32} × 5 – 11 Innermost grouping symbols first

Now you have to calculate everything inside the remaining grouping symbols:

= {4 × 7 + 9} × 5 – 11 first work out the exponent

= {28 + 9} × 5 – 11 then do the multiplication

= 37 × 5 – 11 then do the additionSolution continues…

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Topic1.2.6

Example 3

Order of OperationsOrder of Operations

Solution (continued)

{4(10 – 3) + 32} × 5 – 11

= {4 × 7 + 32} × 5 – 11

= {4 × 7 + 9} × 5 – 11

= {28 + 9} × 5 – 11

= 37 × 5 – 11

Now there are no grouping symbols left, so you can do the rest of the calculation:

= 185 – 11 do the multiplication first

= 174 and finally the subtraction

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Topic1.2.6

Guided Practice

Solution follows…

Order of OperationsOrder of Operations

Evaluate the following:

17. 24 ÷ 8 – 2

18. 32 – 4 × 2

19. 21 – {–3[–5 × 4 + 32] – 9 × 23} + 17

20. –3(32 – 4)

3 + 12 ÷ [10 + 3(–4)]

21. 8 + {10 ÷ [11 – 6] × (–4)}

22. [17 + {(–33 + 4) × 8 – 17}] ÷ 8

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21 – {–3 × –11 – 9 × 23} + 17= 21 – {33 – 72} + 17= 77

8 + {10 ÷ 5 × (–4)} = 8 + {2 × (–4)} = 0

[17 + {–23 × 8 – 17}] ÷ 8= [17 + {–184 – 17}] ÷ 8= –23

–3(9 – 4)

3 + (12 ÷ [–2])

–3 × 5

3 + (–6)= = 5

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Topic1.2.6

Independent Practice

Solution follows…

Order of OperationsOrder of Operations

Evaluate the following:

1. 14 – [9 – 4 × 2]

2. 11 + (9 – 3) – (4 ÷ 2)

3. (–1) × (7 – 10 + 12)

4. 12 + 9 × [(1 + 2) – (6 – 14)]

5. [(11 – 8) + (7 – 2)] × 3 – 13

6. [(10 + 9 + 5) ÷ 2] – (6 – 12)

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15

–9

111

11

18

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Topic1.2.6

Independent Practice

Solution follows…

Order of OperationsOrder of Operations

Insert grouping symbols in each of the following statements so that each statement is true:

7. 12 + 42 × 24 – 18 ÷ 3 = 4412 + 42 × [(24 – 18) ÷ 3] = 44 or 12 + 42 × [(24 – 18) ÷ 3] = 44

8. 20 + 32 – 14 – 12 × 6 = 517 (20 + 3)2 – (14 – 12) × 6 = 517

9. 4 × 33 – 2 × 32 – 3 × 4 + 6 = 1662 (4 × 3)3 – (2 × 3)2 – 3 × (4 + 6) = 1662

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Topic1.2.6

Round UpRound Up

Order of OperationsOrder of Operations

You really need to learn the order of operation rules.

You’ll be using them again and again in Algebra I so you might as well make sure you remember them right now.